Average Error: 30.5 → 16.8
Time: 7.3s
Precision: 64
Internal Precision: 576
\[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -4.923437784376753 \cdot 10^{+98}:\\ \;\;\;\;\log \left(-re\right)\\ \mathbf{if}\;re \le 6.959884185585397 \cdot 10^{-191}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{if}\;re \le 1.6666434564112954 \cdot 10^{-170}:\\ \;\;\;\;\log re\\ \mathbf{if}\;re \le 3.964282394890729 \cdot 10^{+104}:\\ \;\;\;\;\log \left(\sqrt{re \cdot re + im \cdot im}\right)\\ \mathbf{else}:\\ \;\;\;\;\log re\\ \end{array}\]

Error

Bits error versus re

Bits error versus im

Derivation

  1. Split input into 3 regimes

  2. if re < -4.923437784376753e+98

    1. Initial program 50.0

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

    2. Taylor expanded around -inf 8.6

      \[\leadsto \log \color{blue}{\left(-1 \cdot re\right)}\]

    3. Applied simplify8.6

      \[\leadsto \color{blue}{\log \left(-re\right)}\]

    if -4.923437784376753e+98 < re < 6.959884185585397e-191 or 1.6666434564112954e-170 < re < 3.964282394890729e+104

    1. Initial program 20.4

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

    if 6.959884185585397e-191 < re < 1.6666434564112954e-170 or 3.964282394890729e+104 < re

    1. Initial program 48.6

      \[\log \left(\sqrt{re \cdot re + im \cdot im}\right)\]

    2. Taylor expanded around inf 11.6

      \[\leadsto \log \color{blue}{re}\]
  3. Recombined 3 regimes into one program.

Runtime

Time bar (total: 7.3s)Debug logProfile

herbie shell --seed '#(1071246582 2318319007 2683472949 3810440501 3233274817 2724848749)' 
(FPCore (re im)
  :name "math.log/1 on complex, real part"
  (log (sqrt (+ (* re re) (* im im)))))